Quantifying relations and similarities of the meteorological parameters among the weather stations in the Alberta Oil Sands region

Our objective was to quantify the similarity in the meteorological measurements of 17 stations under three weather networks in the Alberta oil sands region. The networks were for climate monitoring under the water quantity program (WQP) and air program, including Meteorological Towers (MT) and Edge Sites (ES). The meteorological parameters were air temperature (AT), relative humidity (RH), solar radiation (SR), barometric pressure (BP), precipitation (PR), and snow depth (SD). Among the various measures implemented for finding correlations in this study, we found that the use of Pearson’s coefficient (r) and absolute average error (AAE) would be sufficient. Also, we applied the percent similarity method upon considering at least 75% of the value in finding the similarity between station pairs. Our results showed that we could optimize the networks by selecting the least number of stations (for each network) to describe the measure-variability in meteorological parameters. We identified that five stations are sufficient for the measurement of AT, one for RH, five for SR, three for BP, seven for PR, and two for SD in the WQP network. For the MT network, six for AT, two for RH, six for SR, and four for PR, and the ES network requires six for AT, three for RH, six for SR, and two for BP. This study could potentially be critical to rationalize/optimize weather networks in the study area.


Introduction
In general, understanding weather conditions (in other words meteorology) of the atmosphere plays a critical role in our sustainable existence on the Earth surface, including real-time weather analysis for forecasting weather-induced calamities [1,2]. Such analysis is based on the commonly monitored meteorological parameters, including air temperature (AT), relative humidity (RH), solar radiation (SR), atmospheric pressure (or barometric pressure, BP), precipitation (PR; i.e., rain, snow, freezing rain, sleet, hail, drizzle, and fog), and wind (i.e., speed and direction) [3]. These parameters may also be used for other applications. For example, the Rationalization of climate monitoring stations have been implementing in Canada over the last three decades, which allowed the reduction of several stations without sacrificing useful details of climatic information [22,23]. However, the approach was often restricted to a maximum of two parameters, such as AT and PR, due to their wider applications in climatic and hydrological modellings [24][25][26][27]. Such an approach is determined by capturing and comparing anomalies in the entire network, and its various subsets. These anomalies were estimated using three approaches, such as: (i) time-series trend analysis per decade [20,21]; (ii) statistical analysis of the parameters, such as, mean, median, variance, standard deviation, and coefficient of variation [24,28,29]; and (iii) spatial descriptive statistics in a GIS (Geographic Information System) environment [27,[30][31][32]. In addition to these approaches, closeness and similarity between two datasets are also estimated using two distinct analyses, i.e., graphical, and quantitative [33]. In graphical analysis, the observations of two stations are visually compared for the same period (time-series plot) for identifying time-related variations between two datasets, such as linear or nonlinear trends, upward and downward shifts [34], and presence of error [33]. Additionally, the scatter plot is another graphical representation that is frequently used to test the model performance (closeness between two datasets) by using the coefficient of determination and slope of the fitted line. On the other hand, the quantitative analysis is classified into two categories, such as (i) analysis of association, and (ii) analysis of coincidence.
Analysis of association is the accuracy estimation using indices, such as coefficient of determination, Pearson correlation coefficient, Spearman's correlation coefficient, Nash-Sutcliffe coefficient, and cosine similarity. Among these indices/measures, Pearson's correlation coefficient is one of the most widely used statistics today, because it determines both the strength and direction of the relationship between two variables [35]. On the other hand, analysis of coincidence includes several metrices, such as absolute average error (AAE), relative difference (RD), mean squared error (MSE), root mean square error (RMSE), and bias (B). Among these, AAE is a more natural measure of average error in compared to the highly used measure of RMSE. This is because, RMSE is a function of three characteristics of a set of errors, while AAE is relatively simple to calculate [36]. While these methods indicate the closeness and similarity between two datasets and the ability of one dataset to predict another, they do not estimate actual similarity, i.e., the number of similar values (data points) in each station-pair in the datasets. Moreover, to the best of our knowledge from literature, we did not find any approach that considered instrumental errors to quantify data closeness or similarity in finding redundant parameters/stations in the weather networks. Hence, assuming the sufficiently dense weather stations, we set our overall goal to rationalize the existing networks with a new concept of percentage similarity (PS) and identify the number of weather stations and their measured parameters that provides a similar set of observations. For this, performing a parametric similarity analysis for the measured parameters, including AT, RH, SR, BP, PR, and snow depth (SD) would be appropriate. In fulfilling our overall goal, the specific objectives in this study were to: i. evaluate various measures in both the analysis of association and coincidence in finding the most representative measures in establishing the relationship; ii. calculate the percentage of similarity of the measurements in the datasets by considering the instrumental errors to find the similarity among the weather stations and meteorological parameters of the networks; and iii. determine an optimal network of weather stations in the oil sands region based on the estimation of the percentage of similarity in each meteorological parameter.

Study area
The lower Athabasca River Basin (ARB) of Northern Alberta in Canada, also a part of the Athabasca oil sands area, is our study area. Three distinct networks of weather stations measure the meteorological parameters in the area (Fig 1)

Meteorological parameters and data availability
Stations C1 to C5 of OSM WQP record daily AT in˚C, RH in %, SR in W/m 2 , BP in kPa, PR in mm, and SD in cm at 2 m height, where L1 and L2 record daily AT, RH, and PR. On the other hand, WBEA MT stations record hourly AT, RH, SR, and PR at 2 m, where AT and RH are also measured at 16, 21, and 29 m. In case of WBEA ES network, all stations record hourly AT, RH, SR, and BP at 2 m. Wind speed and direction is also measured at these stations, but this variable was not analyzed in the scope of this paper. The available period of data records for the networks are provided in Table 2.

Analysis of association
We applied several measures including Pearson correlation coefficient (r), coefficient of determination (R 2 ), Spearman's correlation coefficient (R s ), Nash-Sutcliffe efficiency (E), and Cosine similarity (Cosθ), as shown in Eqs 1 to 5.
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi P n i¼1 ðD1 i À � D1Þ 2 q ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi P n i¼1 ðD2 i À � D2Þ

PLOS ONE
Quantifying relations and similarities of the meteorological parameters among the weather stations Cosy ¼ P n i¼1 D1D2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi P n i¼1 D1 2 q ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi where D1, and D2 refers to observational data recorded at Station A and B, respectively, n is the number of observations, RSS is the residual sum of squares, and TSS is the total sum of squares.

Analysis of coincidence
We adopted several measures including AAE, RD, MSE, RMSE, and B, as shown in Eqs 6 to 10. Note that the meaning of all symbols in these equations are same as described in sub-section 3.1. Analysis of Association.
RMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 n

Determination of the representative measure
It might be possible that one estimate from each of the association and coincidence measures could sufficiently describe the similarities in meteorological observations to represent the entire datasets. Identifying one such measure would reduce the ambiguity in using multiple measures. Therefore, to find such a representative measure in each group, we performed linear regressions among measures. In case of the association, where all measures were dimensionless, we performed the comparison of the estimates from all meteorological parameters together. In contrast for the measures of coincidence, we separately plotted the estimates derived for various meteorological parameters as the estimates were in different units.

Similarity analysis
We performed a similarity analysis on the station pairs for the variables (i.e., meteorological parameters) using acceptable values of the instrumental error suggested by the standard operating procedure (SOP; [40][41][42][43]) by applying Eq 11.
where N1 is the total data count, and N2 is the count that satisfy the following set of arguments: • If the absolute difference between D1 and D2 are ± 0.5˚C for AT, ± 5% for RH, and ± 2.5 cm for SD as suggested in the SOP; and • If the % deviation between D1 and D2 falls 20% for hourly SR, 10% for daily SR, 2% for PR, and 1% for BP as per the SOP. In this case, the deviation is calculated based on the higher value between D1 and D2.

Comparison among the measures of association
Fig 2 shows the plots of association measures that were estimated for all the meteorological parameters with reference to r. Our analyses revealed that the association between two datasets could be described by using only r, as high correlations (R 2 > 0.84) were observed between r and other measures (i.e., R 2 , R s , Cosθ, E) in most cases. The R 2 indicates the proportion of variance in one variable due to another. In general, values of R 2 if higher than 0.50 were considered as significant and acceptable [44,45] and values of R 2 higher than 0.70 were considered as strong [46,47]. In case of E, we considered only the positive estimates as negative estimates would indicate that the observed mean would be a better predictor in this context [48]. Consequently, we opted to use r as a representative for estimating the measures of association for the entire datasets.

Comparison among measures of coincidence
For the comparison, we plotted AAE against other relevant measures (i.e., RMSE, MSE, RD and B) for various meteorological parameters (see Figs 3 and 4). In general, the AAE estimates were well correlated with strong R 2 values (> 0.96) with all other estimates except for RD and B. It would be the case as RD provided an estimate of percent deviation from the highest record between the two stations for a parameter of interest, while AAE, RMSE and MSE provided the actual differences. On the other hand, B presented the summation of both negative and positive differences between two data records; thus it would potentially give different values in comparison to AAE, RMSE and MSE [49]. Taking these into consideration, we assumed to employ AAE as a representative for estimating the measures of coincidence for the entire datasets.

Relations and similarity analysis
We determined the relations and similarity for each station-pair for the associated parameters according to the estimated values of r, AAE, and PS. Here, we considered at least 75% for PS in finding the similarity between station-pairs. Estimated values of r, AAE, and PS for each station-pair of the three networks are shown in Tables 3-6, and associated regression equations are shown in Supporting Information. A detailed relations and similarity analysis for identifying the least number of required station/s in each network for each meteorological parameter are presented in the following sub-sections. Note that the analysis was performed in each network separately (instead of all three networks combined) and we considered it as a limitation in this study. It was because, we wanted to retain the different goals and purposes for establishing three distinct networks by the agencies. Table 3 shows relations and similarity analysis on 21 stationpairs for the six associated meteorological parameters (i.e., AT, RH, SR, BP, PR, and SD) in this network. For AT, we found that all the station-pairs were strongly related (i.e., r � 0.99  Considering the PS values, we found that at least five stations (i.e., C1, C2, C4, C5, and L1) would be required for representing this network. The other two stations (i.e., C3 and L2) were similar to C1 station, which would be likely due to their spatial closeness (see Fig 1) and similar altitude (303 to 331 m msl) [50].   For RH, we observed that all the station-pairs were highly related with a r � 0.83 and AAE values within the acceptable range of SOP (i.e., less than 10% [41]; see Table 3). We also noticed that the PS values in the station-pairs were greater than 80% except the C4 vs C5 and C5 vs L1. Considering the PS values, we identified that only one station (i.e., C1 with the longest data records) would be required for representing this network. Such higher similarity was observed possibly because they were very close to waterbodies [51]. Note that station L2 was giving the highest PS value, however, this station did not record all meteorological parameters.

OSM WQP network.
In case of SR, we observed that all the station-pairs were very strongly related with a r values � 0.92 and low AAE values � 29.35 W/m 2 (see Table 3). However, the PS values were less than or equal to 42.29%, and therefore, we considered that there was no acceptable similarity in related to SR among the stations, and at least five stations (i.e., C1, C2, C3, C4, and C5) would be required in the network. Note that two stations (i.e., L1 and L2) did not record SR measurements. Such dissimilarity would probably be attributed to several factors, including altitude, terrain, air quality, cloud cover, and vegetation that affect the amount of SR received at any place [52]. In case of the parameter BP, we found that four stations (i.e., C2, C3, C4, and C5) were recording the data. These station-pairs showed their strong relationships with a r values � 0.81 and AAE values in the range of 0.48 to 3.17 kPa (see Table 3). Considering the PS values, we identified that at least three stations (i.e., C2, C3, and C5) would be required for representing this network (see Table 3). The other station (i.e., C4) were similar to C3 station (i.e., PS = 98.41%), which would be likely due to their similar altitudes (i.e., 295 and 305 m msl) [53].
Further, in case of PR, we noticed that the station-pairs were having reasonable relations as reflected in r (between 0.46 and 0.83) and AAE values (between 0.56 and 1.47 mm) (see Table 3). However, PS values were extremely low for all station-pairs (i.e., between 2.91 and 9.79%); where such a high dissimilarity in the network was due to the variable nature of PR that was evident even at any small scale [9]. This also suggested that all seven stations (i.e., C1, C2, C3, C4, C5, L1, and L2) would be required for representing PR in this network.
Finally in case of SD, station C1 showed very strong relations (i.e., � 0.92) and least error (i.e., AAE � 3.87 cm) among the four station-pairs (see Table 3). Besides, we found that PS values of the three station-pairs with C1 were strong (i.e., > 76%; see Table 3), except C5 station with PS value of 70.10%. These measures indicated that C1 would be a representative for SD in the network for three stations, i.e., C2, C3, and C4. Here, such strong similarity would be due to having little altitude differences in locations of the stations, which attributed to the fact that snow accumulation on the ground was depended on latitude and time of year in addition to altitude, vegetation and wind [54]. Moreover, dissimilarity of C5 (altitude 559 m msl) with C1 station (altitude 303 m msl) would be due to the same fact of having higher elevation differences of the locations. Therefore, both C1 and C5 stations would be the least required ones for SD in the network. Tables 4 and 5 show the relations and similarity analysis of six stations (i.e., JP104, JP107, JP201, JP213, JP311, and JP316) for AT, RH, SR, and PR parameters, where AT and RH were measured at different heights (i.e., 2, 16, 21, and 29 m; see Table 4). In this network, we found that all station-pairs were highly related with r values from 0.72 to 0.99 and AAE values ranges from 1.46 to 4.5˚C (see Table 4). However, the PS values were significantly low (i.e., between 20.30 to 46.52%) for all station-pairs. Considering the PS values, all stations were required in the network for AT measurements. Such low similarity in the network would be due to the largely spaced distribution of the stations (see Fig 1), and significant altitude differences among these stations (i.e., 256 to 626 m msl) [50].

WBEA MT network.
In the view for RH, all station-pairs showed reasonably strong relations with r values from 0.73 to 0.90 and AAE values in the range of 5.92 to 11.06% (see Table 4). Considering the PS values, we identified that at least two stations (i.e., JP104 and JP201) would be required for representing the RH this network. The other stations (i.e., JP107, JP213, JP311, and JP316) were similar to JP104 station (i.e., PS values in the range of 75.81 to 82.08%). Note that strong similarity for RH in this network was observed, because RH value would be similar over a region of interest [55].
Next, in case of SR, we found that all station-pairs in the network were strongly related with the r values in the range of 0.85 to 0.93 and low AAE values ranges from 35.98 to 63.39 W/m 2 , except the station-pair JP104 vs JP201 with r = 0.18 and AAE = 249.04 W/m 2 (see Table 5). However, we noticed that PS values were significantly low (i.e., � 45.83%) for all station-pairs, and therefore, all stations (i.e., JP104, JP107, JP201, JP213, JP311, and JP316) were required for the measurements of SR in the network. Such significant dissimilarity in SR was attributed to the factors including varying topography (i.e., high differences in the station-altitudes from 256 m to 626 m msl), and cloud cover that might affected the amount of SR received [52].
Lastly for PR, we found that four stations (i.e., JP107, JP213, JP311, and JP316) were recording the data. Here, all stations-pairs were having weak relations with r values � 0.31 and AAE values in the range of 0.55 to 0.66 mm (see Table 5). In addition, we observed that the PS values were extremely low (i.e., � 2.73%), and therefore, all stations were required for the measurements of PR in the network. Such dissimilarity was likely due to the variable nature of PR observed even in a short distances [9].

WBEA ES network.
We presented relations and similarity analysis for the six stations (i.e., JE306, JE308, JE312, JE316, JE323, and R2) of WBEA ES network in Table 6. For AT measurements, we found that all station-pairs were having very strong relations with r values ranges from 0.97 to 0.99 and acceptable AAE values in the range of 1.53 to 2.66˚C (see Table 6) [56]. However, the PS values were low, i.e., in the range of 26.41 to 49.72%. Considering the PS values, all six stations were required in the network. Here, such a dissimilarity was observed probably due to the widely spaced distribution of the stations (see Fig 1), and significant altitude differences [50] that ranged from 299 m to 520 m msl.
In case of RH, we identified that all station-pairs were having very strong relations with the values of r ranges from 0.82 to 0.92 and AAE in the range of 5.77 to 8.66% except one stationpair (JE312 vs R2) having a reasonable relation (i.e., r = 0.65 and AAE = 11.67%; see Table 6). Here, the PS values were in the range of 61.42 to 84.26%. Considering the PS values, we identified that JE306, JE308 and JE323 stations were required for the measurements of RH in the network. Overall, the similarity of the stations in this network was likely related to having similar landcover [55].
While comparing station-pairs for the measurements of SR, we observed that all were showing strong relations with the r values ranges from 0.76 to 0.90 and AAE values in the range of 53.04 to 92.06 W/m 2 (see Table 6). However, in all cases, we noticed that PS values were considerably low (i.e., � 42.54%). Considering the PS values, we identified that all stations were necessary for the measurements of SR in the network. Such a low similarity would be due to the several factors, including varying topography and cloud cover that could affect receiving amount of SR over a place [52].
In the event of BP, we noticed that the relations were weak to moderate (i.e., r values range from 0.11 to 0.55 and AAE values in the range of 0.12 to 2.56 kPa) for all station-pairs in the network (see Table 6). Here, the PS values covered a wide range, i.e., 0 to 99.99%. Considering the PS values, we found that two stations (i.e., JE306 and JE312) were required for measuring BP in the network. Here, strong similarities were likely due to their locations in the valley region with similar altitudes [50].

Suggested optimization
We synthesized the network-specific required weather stations for each of the meteorological parameters (see Table 7) based on similarity analysis detailed in Section 3.4 and 4.3. We observed that all the stations were required for some of the parameters, i.e., (i) PR for OSM WQP, (ii) AT, SR, and PR for WBEA MT; and (iii) AT, and SR for WBEA ES networks. Though, each of the networks might be optimized for some of the parameters but not for all; thus, we would require keeping all the existing stations. As these would be cases, then we would consider not to remove the parameter-specific sensors that even showed redundancy. It would be due to their useability in case of failure of similar sensors in other stations.

Conclusions
In this study, we demonstrated that the PS analysis could quantify similarity between weather stations. It determines the least number of station/s to fully represent the spatial variability in climate measurements required in a network of interest. Moreover, such similarity analysis is a better measure in compared to the relational measures like r and AAE in quantifying the similarity between two meteorological datasets. It also assists to identify the best representative station for a particular parameter in a network that could represent the area. In most of the instances, we noticed that the station-pairs were related. However, in case of similarity, we identified that the measurements from five, three, two, and one station/s would be the least required for the measurements of AT, BP, SD, and RH parameters, respectively in the OSM WQP network. For the same network, we also found that seven and five stations were the least required ones for the measurements of SR and PR, respectively. For the WBEA MT network, we identified that all six stations were required for the measurements of AT, SR, and PR, where two would be sufficient for RH. Moreover, in case of WBEA ES network, we found that the measurements from all stations were the least required ones for AT and SR parameters. However, only three stations would be adequate for RH, and two for BP, in this network. Note that we could not perform similarity analysis on few station-pairs, because some stations did not record some specific meteorological parameters of our interest. Nevertheless, we showed that the similarity analysis of using the PS value had potential applications to rationalize/optimize weather station network (stations and parameters) in the study area. It would help to minimize the associated operational costs without sacrificing the scientific credibility of the monitoring programs. However, we recommend evaluating these methods thoroughly before applying them to other weather networks in Canada, and elsewhere during any decision-making process. Further, apart from meteorological study, PS could be a tool to find similarity between two datasets with same parameter in other field of research. Supporting information S1